Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 34

1/12-octave filters are obtained by dividing each one-octave band into 12 geometrically equal sub‐sections, i.e. f_U/f_L = 2^1/12. By the same procedure as for octave and one‐third‐octave filters, we get the result that for 1/12‐octave bands the cutoff frequencies, f_L and f_U, are f_C × 2^−1/24 and f_C × 2^1/24, respectively. Then, the bandwidth is given by Δf = f_U − f_L = f_C(2^1/24 − 2^−1/24), so Δf ≈ 6% (f_C).

There are two main types of constant percentage filters in common use: (i) those with a fixed center frequency and bandwidth which is a certain percentage of the center frequency, and (ii) those with a variable (or tunable) center frequency and a bandwidth which can be set to certain selected percentages of the center frequency. The first type of filter is perhaps in most common use. In practical instruments, many different parallel filters each with a different center frequency are assembled in one unit. The instrument is provided with a root mean square detector and a display.

On the other hand, instruments for constant bandwidth filter analysis are normally constructed so that the center frequency of a single filter can be tracked effectively throughout the frequency range of interest. Often different bandwidth settings are available on the same instrument (e.g. 1, 5, 10, 20 Hz). The narrower the bandwidth chosen, the slower the tracking rate should be to obtain reliable results. A rule which should be used in spectrum analysis is that the duration, T, of the noise sample length (or of the analysis time) must be at least as long as the reciprocal of the bandwidth Δf,

(1.16)

This fundamental principle, also known as the uncertainty principle, puts a limit on the corresponding resolutions in the time and frequency domain, meaning that narrow resolution in one domain means wide resolution in the other domain [10].

If fine frequency resolution is not needed or if the signal is broadband in nature, then octave band readings are sufficient. One‐third octave band readings (or narrower) should be used if the signal spectrum is not smooth or if pure tones are present. For diagnostic work on machinery, it may be necessary to use constant bandwidth filters (e.g. if the fan blade passing frequency and its higher harmonics must be separated).

When changes in level and frequency of a signal occur in a short period of time, real‐time frequency analysis is required to observe rapid variations in the signal and showing the results on a continuously updated display. Real‐time analysis can be performed using a frequency analyzer made up of a set of parallel filters and a detector (see Figure 1.12). The input signal is previously conditioned in terms of level (gain/attenuation) and high‐ and/or low‐pass filtering. A digital analyzer will require an anti‐aliasing filter at the input before the ADC. The conditioner is then connected to a large number of parallel band‐pass filter channels (usually between 30 and 40 for a standard one‐third octave band model). The detector detects the power in the transmitted signal in terms of its mean square or rms value. In Figure 1.12, only one detector is shown, and this is supposed to work as detector for all the filters in the situation of a parallel filter bank [10].

Figure 1.11 Comparison between bandwidths of (a) constant percentage and (b) constant bandwidth filters at the same frequency.

Figure 1.12 Simplified block diagram of a parallel‐filter real‐time analyzer [10].